DLMF:22.6.E21 (Q6955): Difference between revisions

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Property / Symbols used
 
Property / Symbols used: Q11985 / rank
 
Normal rank
Property / Symbols used: Q11985 / qualifier
 
Defining formula:

z 𝑧 {\displaystyle{\displaystyle z}}

z
Property / Symbols used: Q11985 / qualifier
 
xml-id: C22.S1.XMD3.m1tdec
Property / Symbols used
 
Property / Symbols used: Q11984 / rank
 
Normal rank
Property / Symbols used: Q11984 / qualifier
 
Defining formula:

k 𝑘 {\displaystyle{\displaystyle k}}

k
Property / Symbols used: Q11984 / qualifier
 
xml-id: C22.S1.XMD4.m1tdec
Property / Symbols used
 
Property / Symbols used: Q11988 / rank
 
Normal rank
Property / Symbols used: Q11988 / qualifier
 
Defining formula:

k ′ superscript 𝑘 ′ {\displaystyle{\displaystyle k^{\prime}}}

k^{\prime}
Property / Symbols used: Q11988 / qualifier
 
xml-id: C22.S1.XMD5.m1ndec

Latest revision as of 15:00, 2 January 2020

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DLMF:22.6.E21
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    Statements

    Defining formula
    dn 2 ⁡ ( 1 2 ⁢ z , k ) = k 2 ⁢ cn ⁡ ( z , k ) + dn ⁡ ( z , k ) + k ′ 2 1 + dn ⁡ ( z , k ) = k ′ 2 ⁢ ( 1 - cn ⁡ ( z , k ) ) dn ⁡ ( z , k ) - cn ⁡ ( z , k ) = k ′ 2 ⁢ ( 1 + dn ⁡ ( z , k ) ) k ′ 2 + dn ⁡ ( z , k ) - k 2 ⁢ cn ⁡ ( z , k ) . Jacobi-elliptic-dn 2 1 2 𝑧 𝑘 superscript 𝑘 2 Jacobi-elliptic-cn 𝑧 𝑘 Jacobi-elliptic-dn 𝑧 𝑘 superscript superscript 𝑘 ′ 2 1 Jacobi-elliptic-dn 𝑧 𝑘 superscript superscript 𝑘 ′ 2 1 Jacobi-elliptic-cn 𝑧 𝑘 Jacobi-elliptic-dn 𝑧 𝑘 Jacobi-elliptic-cn 𝑧 𝑘 superscript superscript 𝑘 ′ 2 1 Jacobi-elliptic-dn 𝑧 𝑘 superscript superscript 𝑘 ′ 2 Jacobi-elliptic-dn 𝑧 𝑘 superscript 𝑘 2 Jacobi-elliptic-cn 𝑧 𝑘 {\displaystyle{\displaystyle{\operatorname{dn}^{2}}\left(\tfrac{1}{2}z,k\right% )=\frac{k^{2}\operatorname{cn}\left(z,k\right)+\operatorname{dn}\left(z,k% \right)+{k^{\prime}}^{2}}{1+\operatorname{dn}\left(z,k\right)}=\frac{{k^{% \prime}}^{2}(1-\operatorname{cn}\left(z,k\right))}{\operatorname{dn}\left(z,k% \right)-\operatorname{cn}\left(z,k\right)}=\frac{{k^{\prime}}^{2}(1+% \operatorname{dn}\left(z,k\right))}{{k^{\prime}}^{2}+\operatorname{dn}\left(z,% k\right)-k^{2}\operatorname{cn}\left(z,k\right)}.}}
    0 references
    DLMF id
    DLMF:22.6.E21
    0 references
    Symbols used
    Jacobian elliptic function
    Defining formula
    cn ⁡ ( z , k ) Jacobi-elliptic-cn 𝑧 𝑘 {\displaystyle{\displaystyle\operatorname{cn}\left(\NVar{z},\NVar{k}\right)}}
    xml-id
    C22.S2.E5.m2ahdec
    0 references
    Jacobian elliptic function
    Defining formula
    dn ⁡ ( z , k ) Jacobi-elliptic-dn 𝑧 𝑘 {\displaystyle{\displaystyle\operatorname{dn}\left(\NVar{z},\NVar{k}\right)}}
    xml-id
    C22.S2.E6.m2ahdec
    0 references
    Q11985
    Defining formula
    z 𝑧 {\displaystyle{\displaystyle z}}
    xml-id
    C22.S1.XMD3.m1tdec
    0 references
    Q11984
    Defining formula
    k 𝑘 {\displaystyle{\displaystyle k}}
    xml-id
    C22.S1.XMD4.m1tdec
    0 references
    Q11988
    Defining formula
    k ′ superscript 𝑘 ′ {\displaystyle{\displaystyle k^{\prime}}}
    xml-id
    C22.S1.XMD5.m1ndec
    0 references
     
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